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a special case of partial integration


In determining the antiderivative of a transcendental (http://planetmath.org/AlgebraicFunction) functionMathworldPlanetmath U whose derivative Uβ€² is algebraic (http://planetmath.org/AlgebraicFunction), the result can be obtained when choosing in the formula

∫UV′𝑑x=UV-∫VU′𝑑x

of integration by parts  V′≑1;  then one has

∫U𝑑x=∫Uβ‹…1𝑑x=Uβ‹…x-∫xβ‹…U′𝑑x.

The functions U in question are mainly the logarithmMathworldPlanetmath (http://planetmath.org/NaturalLogarithm2), the cyclometric functions and the area functions.

Examples.

  1. 1.

    ∫lnxdx=xlnx-∫xβ‹…1x𝑑x=xlnx-x+C

  2. 2.

    ∫arcsinxdx=xarcsinx-∫xβ‹…1√1-x2𝑑x=xarcsinx+12∫-2x√1-x2𝑑x=xarcsinx+√1-x2+C

  3. 3.

    ∫arctanxdx=xarctanx-∫xβ‹…11+x2𝑑x=xarctanx-12∫2x1+x2𝑑x=xarctanx-12ln(1+x2)+C=xarctanx-ln√1+x2+C

  4. 4.

    ∫arcoshxdx=xarcoshx-∫xβ‹…1√x2-1𝑑x=xarcoshx-√x2-1+C

The choice  V′≑1  works as well in such cases as  ∫(lnx)2𝑑x  and  ∫ln(lnx)𝑑x,  giving respectively  x((lnx)2-2lnx+2)+C  and  xln(lnx)-Lix+C (see logarithmic integralDlmfDlmfMathworldPlanetmathPlanetmath). Also  ∫(arcsinx)2𝑑x  , requiring two integrations by parts, and giving the result  x(arcsinx)2+2√1-x2arcsinx-2x+C.

Title a special case of partial integration
Canonical name ASpecialCaseOfPartialIntegration
Date of creation 2013-03-22 17:38:35
Last modified on 2013-03-22 17:38:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Feature
Classification msc 26A36
Related topic IntegralTables